I have AVG antivirus Free Edition.

When I run a scan, get a report, and click on "Warnings tab, then hit "Remove all Unheald Infections", the computer locks up.

The mouse pointer changes into an hourglass, and just sits there frozen. I'm not a very patient person. There is no indication that any activity is occuring, so this became unbearably frustrating after five minutes, and I shut AVG down.

What is going on? What is the problem?

I would imagine that you have A LOT of warning/infections. When you click heal your processor may be slow and its trying to heal a large amount of infections and your processor is just lagging while trying to heal them. Give it more time or Do the following.

Reinstall see if that helps.
Upgrade the processor/RAM <- might be a good idea anyways for overall performance.

1.7GHz and 1.5GIG of RAM or better for good performance. That minimum i would think.